Quantum Heegaard diagrams and knot Floer Homology

TL;DR

This paper introduces quantum Heegaard diagrams that unify the Alexander and Jones polynomials through a geometric intersection model, providing new insights into knot invariants and their categorifications.

Contribution

It constructs quantum Heegaard diagrams from braid presentations, linking knot Floer homology with a new geometric framework that recovers both Alexander and Jones polynomials.

Findings

Quantum Heegaard diagrams unify Alexander and Jones polynomials.

Knot Floer homology is viewed as a categorification of the Alexander polynomial.

A new geometric intersection formula for the Jones polynomial is proposed.

Abstract

Given a knot presented as a braid closure, we construct a unified intersection model for the Alexander and Jones polynomials of the knot via what we call quantum Heegaard diagrams. These diagrams are obtained by stabilising the disc model of the first author, which we show are doubly-pointed Heegaard diagrams of the knot together with an additional set of base points. We identify the Alexander grading in the disc model with the Alexander grading in the Heegaard diagram. As the Lagrangian intersection Floer homology of the Heegaard tori in the symmetric power of the Heegaard surface is knot Floer homology, we can view knot Floer homology as a natural categorification of the Alexander polynomial arising from the disc model. The additional base points let us define a new grading on the intersection between the Heegaard tori, which we call quantum Alexander grading. Combining this with the classical Alexander grading, we define a two-variable graded intersection between the Heegaard tori that recovers the Jones and Alexander polynomials as two specialisations of coefficients. The resulting intersection formula for the Jones polynomial opens up a potential avenue to obtaining a new geometric categorification of the Jones polynomial.